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Degree 2 Bernstein on a triangle
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- \(R\) is the reference triangle. The following numbering of the subentities of the reference is used:
- \(\mathcal{V}\) is spanned by: \(1\), \(x\), \(x^{2}\), \(y\), \(x y\), \(y^{2}\)
- \(\mathcal{L}=\{l_0,...,l_{5}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(0,0)\)
\(\displaystyle \phi_{0} = x^{2} + 2 x y - 2 x + y^{2} - 2 y + 1\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \phi_{0} = x^{2} + 2 x y - 2 x + y^{2} - 2 y + 1\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{1}:v\mapsto v(1,0)\)
\(\displaystyle \phi_{1} = x^{2}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \phi_{1} = x^{2}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{2}:v\mapsto v(0,1)\)
\(\displaystyle \phi_{2} = y^{2}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \phi_{2} = y^{2}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{3}:v\mapsto c^{e_{0}}_{1}\)
where \(v=\sum_ic^{e_{0}}_iB^{e_{0}}_i\);
\(B^{e_{0}}_1\) to \(B^{e_{0}}_n\) are the degree 2 Bernstein polynomials on \(e_{0}\);
and \(e_{0}\) is the 0th edge.
\(\displaystyle \phi_{3} = 2 x y\)
This DOF is associated with edge 0 of the reference element.
where \(v=\sum_ic^{e_{0}}_iB^{e_{0}}_i\);
\(B^{e_{0}}_1\) to \(B^{e_{0}}_n\) are the degree 2 Bernstein polynomials on \(e_{0}\);
and \(e_{0}\) is the 0th edge.
\(\displaystyle \phi_{3} = 2 x y\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{4}:v\mapsto c^{e_{1}}_{1}\)
where \(v=\sum_ic^{e_{1}}_iB^{e_{1}}_i\);
\(B^{e_{1}}_1\) to \(B^{e_{1}}_n\) are the degree 2 Bernstein polynomials on \(e_{1}\);
and \(e_{1}\) is the 1st edge.
\(\displaystyle \phi_{4} = 2 y \left(- x - y + 1\right)\)
This DOF is associated with edge 1 of the reference element.
where \(v=\sum_ic^{e_{1}}_iB^{e_{1}}_i\);
\(B^{e_{1}}_1\) to \(B^{e_{1}}_n\) are the degree 2 Bernstein polynomials on \(e_{1}\);
and \(e_{1}\) is the 1st edge.
\(\displaystyle \phi_{4} = 2 y \left(- x - y + 1\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{5}:v\mapsto c^{e_{2}}_{1}\)
where \(v=\sum_ic^{e_{2}}_iB^{e_{2}}_i\);
\(B^{e_{2}}_1\) to \(B^{e_{2}}_n\) are the degree 2 Bernstein polynomials on \(e_{2}\);
and \(e_{2}\) is the 2nd edge.
\(\displaystyle \phi_{5} = 2 x \left(- x - y + 1\right)\)
This DOF is associated with edge 2 of the reference element.
where \(v=\sum_ic^{e_{2}}_iB^{e_{2}}_i\);
\(B^{e_{2}}_1\) to \(B^{e_{2}}_n\) are the degree 2 Bernstein polynomials on \(e_{2}\);
and \(e_{2}\) is the 2nd edge.
\(\displaystyle \phi_{5} = 2 x \left(- x - y + 1\right)\)
This DOF is associated with edge 2 of the reference element.